Problem: $\overline{BC} = 4$ $\overline{AC} = {?}$ $A$ $C$ $B$ $?$ $4$ $ \sin( \angle ABC ) = \frac{7\sqrt{65} }{65}, \cos( \angle ABC ) = \frac{4\sqrt{65} }{65}, \tan( \angle ABC ) = \dfrac{7}{4}$
Solution: $\overline{AC}$ is the opposite to $\angle ABC$ $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the adjacent side and need to solve for the opposite side so we can use the tan function (TOA) $ \tan( \angle ABC ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\overline{AC}}{\overline{BC}}= \frac{\overline{AC}}{4} $ $ \overline{AC}=4 \cdot \tan( \angle ABC ) = 4 \cdot \dfrac{7}{4} = 7$